Eu gosto de dança.

Because the verb gostar normally requires the preposition de before what you like.

• gostar de + noun
  • Eu gosto de dança.
  • Ela gosta de música.
• gostar de + infinitive
  • Eu gosto de dançar.
  • Eles gostam de viajar.

So Eu gosto dança would be incorrect.

Gosto is the 1st person singular present-tense form of gostar.

So:

• eu gosto = I like
• você gosta = you like
• ele/ela gosta = he/she likes

In this sentence, gosto matches eu.

Yes. In Portuguese, subject pronouns are often omitted when the verb already makes the subject clear.

So both of these are possible:

• Eu gosto de dança.
• Gosto de dança.

The version with eu can sound a bit more explicit or emphatic, but both are natural.

Here, dança is a noun, so it means dance as a thing, field, or art form.

• dança = dance
• dançar = to dance

That means there is an important difference:

• Eu gosto de dança. = I like dance / I like dance as an art form or subject.
• Eu gosto de dançar. = I like dancing / I like to dance.

Usually, no.

If you mean I enjoy the activity of dancing, the more natural sentence is:

• Eu gosto de dançar.

Eu gosto de dança is more likely to mean that you like dance in a general sense, like an artistic area, similar to saying you like music, theater, or art.

So:

• activity: Eu gosto de dançar.
• art form / subject: Eu gosto de dança.

Yes, article use can change the nuance.

• de dança = dance in a general, broad sense
• da dança = the dance / dance as something more specific or more definite in context

For a broad category, Portuguese often allows no article:

• gosto de música
• gosto de teatro
• gosto de dança

If you are talking about something more specific, an article is more likely:

• Eu gosto da dança contemporânea.
• Eu gosto da dança que eles apresentaram.

So Eu gosto de dança is general, while Eu gosto da dança usually sounds more specific.

A simple Brazilian Portuguese approximation is:

eh-ooh GOS-too djee DAHN-sah

A few helpful notes:

• Eu sounds roughly like eh-ooh said quickly together.
• gosto has stress on GOS.
• In many Brazilian accents, de sounds like djee.
• dança has stress on dan.

A more careful approximate breakdown:

• Eu → eh-ooh
• gosto → GOS-too
• de → djee
• dança → DAHN-sah

Yes. Dança is a feminine noun: a dança.

That matters when you use articles and adjectives:

• a dança
• a dança brasileira
• a dança clássica

In your sentence, you do not see the article, but the noun is still feminine.